Suppose yn represents the population of the world the nth year after 1 800. That is. y0 is the population in 1800, y1 is the population in 1801 and so on. From the definition of the derivative as the limit of a difference quotient, we can write

So Malthus's differential equation model dy/dt = 0.03y can be approximated by y(t + 1) - y(t) = 0.03y. that is. y(t + 1) = y(t) + 0.03y(t ). Thus writing yn for y(n) we obtain
yn + 1 = l .03yn·
(a) Use this discrete model (5 ) to estimate the population in the years 1801, 1802. 1803, . . . 1810.
(b) Estimate the world's population in the year 1 900using this discrete model. You might do this with a spreadsheet, using equation (5), or you might develop an algebraic formula by observing a pattern.
(c) Comment on the difference in results comparing this discrete process with Malthus's continuous model in Table 1.1.1 Explain.

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=y(t + 1)-y(t) Table 1.1.1 Comparison of Malthus model y(t) -0.9e0.03 and actual world population (in billions). Year t Malthus Actual YeartMalthos Actnal 1800 181010 1820 20 183030 1840 40 185050 186060 187070 1880 80 1890 90 13.40 190010018.09 0.9 24.42 32.98 44.52 0 0.90 1.21 1.64 2.21 2.99 4.03 5.45 7.35 9.93 1910 110 1920 120 1930 130 1940140 60.10 1950150 1960160 109.53 1970 170 147.87 1980180199.62 1990190 269.49 2000 200 363.81 1.8 0.9 2.3 2.7 3.0 3.5 4.2 5.1 6.0 1.0 81.13 1.2 1.5 1.7